2018 AMC 10A Problems/Problem 16
Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?
As the problem has no diagram, we draw a diagram. The hypotenuse has length . Let be the foot of the altitude from to . Note that is the shortest possible length of any segment. Writing the area of the triangle in two ways, we can solve for , which is between and .
Let the line segment be , with on . As you move along the hypotenuse from to , the length of strictly decreases, hitting all the integer values from (IVT). Similarly, moving from to hits all the integer values from . This is a total of distinct line segments. (asymptote diagram added by elements2015)
Solution 2 - Circles
Note that if a circle with an integer radius centered at vertex intersects hypotenuse , the lines drawn from to the points of intersection are integer lengths. As in the previous solution, the shortest distance . As a result, a circle of will not reach the hypotenuse and thus does not intersect it. We also know that a circle of radius intersects the hypotenuse once and a circle of radius intersects the hypotenuse twice. Quick graphical thinking or Euclidean construction will prove this.
It follows that we can draw circles of radii and that each contribute two integer lengths (since these circles intersect the hypotenuse twice) from to and one circle of radius that contributes only one such segment. Our answer is then ~samrocksnature
Solution 3 - Triangle Inequality
Video Solution 1
Video Solution 2
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